Oxen were very expensive in biblical times. Few farmers owned even a single team of them. In ancient documents from elsewhere in the Near East, there are records of farmers renting them from wealthy owners or even government officials.
My assumption that that Elisha owns both the oxen and the land is that he's in charge of the whole team of and yet he's driving a plow himself. If he was just a farm hand, the text wouldn't have made an issue of him also driving a team. And the fact that he sacrificed two of the oxen strongly implies ownership - otherwise it would have been quite a crime, because oxen were very expensive.
I read the text as telling readers that Elisha owns 12 pairs of oxen in order to inform them that he was an industrious farmer and a man of means. But Elisha was also impassioned about his calling as Elijah's successor, because he burned his plow and sacrificed a pair of oxen, showing his commitment to leave behind his prosperous agrarian lifestyle. He asks to kiss his parents goodbye, likely wanting to have his family's blessing. But when Elijah challenges him, he forgoes even that. This tells you that he's committed to family but willing to leave absolutely everything behind to follow Elijah. These details are provided to show you that Elisha was the right man to be the successor to Elijah, God's fiery, impassioned prophet.
OT narratives often select a few significant details that give key information about a character rather than describing him or her in many flowery words. That's partly because Hebrew doesn't have a lot of adjectives, and partly the style of the text was to "show" rather than "tell."